Problem: Write the equation for a parabola with a focus at $(-4,-2)$ and a directrix at $y=3$. $y=$
Solution: The strategy A parabola is defined as the set of all points that are the same distance away from a point (the focus) and a line (the directrix). Let $(x,y)$ be a point on the parabola. Then the distance between $(x,y)$ and the focus, $(-4,-2)$, is equal to the distance between $(x,y)$ and the directrix, $y=3$. Once we find these distances, we can equate them in order to derive the equation of our parabola. Finding the distances from $(x,y)$ to the focus and the directrix The distance between $(x,y)$ and $(-4,-2)$ is $\sqrt{(x+4)^2+(y+2)^2}$. [How did we find that?] Similarly, the distance between $(x,y)$ and the line $y=3$ is $\sqrt{(y-3)^2}$. [How did we know that?] Deriving the formula by equating the distances $\begin{aligned} \sqrt{(y-3)^2} &= \sqrt{(x+4)^2+(y+2)^2} \\\\ (y-3)^2 &= (x+4)^2+(y+2)^2 \\\\ {y^2}-6y{+9} &= (x+4)^2{+y^2}{+4y}+4\\\\ -6y{-4y}&=(x+4)^2+4{-9} \\\\ -10y&=(x+4)^2-5 \\\\ y&=-\dfrac{(x+4)^2}{10}+\dfrac{1}{2}\end{aligned}$ The answer The equation of our parabola is $y=-\dfrac{(x+4)^2}{10}+\dfrac{1}{2}$. Here is the graph of our parabola. As expected, the distance between a point on the parabola, $(x,y)$, and the focus is the same as the distance between $(x,y)$ and the directrix. ${2}$ ${4}$ ${6}$ ${8}$ ${10}$ ${12}$ ${14}$ ${\llap{-}4}$ ${\llap{-}6}$ ${\llap{-}8}$ ${\llap{-}10}$ ${\llap{-}12}$ ${\llap{-}14}$ ${2}$ ${4}$ ${6}$ ${8}$ ${10}$ ${12}$ ${14}$ ${\llap{-}4}$ ${\llap{-}6}$ ${\llap{-}8}$ ${\llap{-}10}$ ${\llap{-}12}$ ${\llap{-}14}$ $y$ $x$ ${(x,y)}$